The work has not been graded but I like the output that was submitted to me. Is it possible for the same prof to do the next assignment I will be submitting? If possible, I will greatly appreciate it.
Problem 1 (20 pts). Normal distributions are used to approximate the sampling distribution of the sample proportion pˆ when sample size is large (i.e., when np ≥ 10, n(1 − p) ≥ 10). Similarly, we can use a normal distribution to approximate a Binomial distribution when sample size is large, as shown in the lecture notes. However, the accuracies of these approximations are unclear (i.e., how close are the approximate answers to the exact answers if exact answers are available)? Here, you are asked to evaluate the accuracies of the normal approximations in several situations, based on a random variable X which follows the Binomial distribution B(n,p). You can easily obtain exact binomial probabilities from software R with command “pbinom” (some books also give exact binomial probability table. Calculating the exact binomial probabilities by hand is also feasible, though a bit tedious). For the following questions, you can obtain exact binomial probabilities using any of these approaches.
(a) (12 pts). Compute the following probabilities using both the Binomial distribution X ∼ B(n, p) (which gives an exact answer) and its normal approximation X ∼ N (np, np(1 − p)) (which gives an approximate answer), and compare the two answers to see if they are close or not:
(i) choose n = 10 and p = 0.2, and then compute P (X ≤ 3) using both methods; (ii) choose n = 10 and p = 0.4, and then compute P (X ≤ 3) using both methods; (iii) choose n = 50 and p = 0.2, and then compute P (X ≤ 8) using both methods; (iv) choose n = 50 and p = 0.4, and then compute P (X ≤ 8) using both methods;
Summarize the above results in a table and state your conclusions in no more than three sentences (i.e., in what cases the normal approximations seem most accurate – the exact answers and approx- imate answers are closest). (b) (8 pts). The sample proportion pˆ can be defined as pˆ = X/n, where X ∼ B(n, p). For the four cases (i)–(iv) in question (a), compute the probabilities P(pˆ ≤ 0.3) using normal approximations (i.e., pˆ ∼ N (p, p(1 − p)/n)). Intuitively (or based on what you have observed in (a)), which ap- proximation(s) do you think may be the most accurate one? (Note: the sample proportion pˆ has no exact distribution available, so we have to use normal approximations.)
(a) Exact and normal approximationsR-Output1) Exact> pbinom(3,10,0.2)  0.8791261 > pbinom(3,10,0.4)  0.3822806 > pbinom(8,50,0.2)  0.3073316 > pbinom(8,50,0.4) …
Delivering a high-quality product at a reasonable price is not enough anymore.
That’s why we have developed 5 beneficial guarantees that will make your experience with our service enjoyable, easy, and safe.
You have to be 100% sure of the quality of your product to give a money-back guarantee. This describes us perfectly. Make sure that this guarantee is totally transparent.Read more
Each paper is composed from scratch, according to your instructions. It is then checked by our plagiarism-detection software. There is no gap where plagiarism could squeeze in.Read more
Thanks to our free revisions, there is no way for you to be unsatisfied. We will work on your paper until you are completely happy with the result.Read more
Your email is safe, as we store it according to international data protection rules. Your bank details are secure, as we use only reliable payment systems.Read more
By sending us your money, you buy the service we provide. Check out our terms and conditions if you prefer business talks to be laid out in official language.Read more